formulae for adding to cosines

[eee]

I found the formulae for adding a sin and a cos, but could not find the formulae for adding a cos and a cos. Adding a cos and a sin is: cos phi = A/(sqrt A**2 + B**2) where cos theta + sin theta = sq(A**2 + B**2) * cost (theta + phi).

I have the following problem: 50cos(500t + 10) + 50cos(500t + 100) the answer is :70.7cos(500t + 55). What is the formulae for finding phi. I know how to get the 70.7.

 

[royhaas]

Use cos(A)+cos(B)=2cos((A+B)/2)cos((A-B)/2).

 

[soroban]

A few more identities


. . . . Sum-to-Product Identities

. . \(\displaystyle \sin A + \sin B \;=\; 2\sin\left(\tfrac{A+B}{2}\right)\cos\left(\tfrac{A-B}{2}\right)\)

. . \(\displaystyle \sin A - \sin B \;=\;2\cos\left(\tfrac{A+B}{2}\right)\sin\left(\tfrac{A-B}{2}\right)\)

. . \(\displaystyle \cos A + \cos B \;=\;2\cos\left(\tfrac{A+B}{2}\right)\cos\left(\tfrac{A-B}{2}\right)\)

. . \(\displaystyle \cos A - \cos B \;=\;\text{-}2\sin\left(\tfrac{A+B}{2}\right)\sin\left(\tfrac{A-B }{2}\right)\)



, , , . . , Product-to-Sum Identites

. . \(\displaystyle \sin A\cdot\sin B \;=\;\tfrac{1}{2}\bigg[\cos(A-B) - \cos(A+B)\bigg]\)

. . \(\displaystyle \sin A\cdot\cos B \;=\;\tfrac{1}{2}\bigg[\sin(A-B) + \sin(A+B)\bigg]\)

. . \(\displaystyle \cos A\cdot\cos B \;=\;\tfrac{1}{2}\bigg[\cos(A-B) + \cos(A+B)\bigg]\)

. . \(\displaystyle \cos A\cdot\sin B \;=\;\tfrac{1}{2}\bigg[\sin(A+B) - \sin(A-B)\bigg]\)